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<p>We prove a parabolically scale-invariant variation of the planarity estimate in [Naff, K.: A planarity estimate for pinched solutions of mean curvature flow. Duke Math. J. <b>171</b>(2), 443–482 (2022) ] for higher codimension mean curvature flow, borrowing ideas from work of Brendle–Huisken–Sinestrari [Brendle, S., Huisken, G., Sinestrari, C.: Ancient solutions to the Ricci flow with pinched curvature. Duke Math. J. <b>158</b>(3), 537–551 (2011) ]. Additionally, we prove convexity for pinched complete ancient solutions of the mean curvature flow in codimension one. Then we put these estimates together to characterize certain pinched complete ancient solutions and shrinkers in higher codimension. We include some discussion of future research directions in this area of mean curvature flow.</p>

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Planarity and convexity for pinched ancient solutions of mean curvature flow

  • 期刊论文

  • 发表日期：

    2026年5月16日

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  • Tang-Kai Lee,
  • Keaton Naff,
  • Jingze Zhu

摘要

We prove a parabolically scale-invariant variation of the planarity estimate in [Naff, K.: A planarity estimate for pinched solutions of mean curvature flow. Duke Math. J. 171(2), 443–482 (2022) ] for higher codimension mean curvature flow, borrowing ideas from work of Brendle–Huisken–Sinestrari [Brendle, S., Huisken, G., Sinestrari, C.: Ancient solutions to the Ricci flow with pinched curvature. Duke Math. J. 158(3), 537–551 (2011) ]. Additionally, we prove convexity for pinched complete ancient solutions of the mean curvature flow in codimension one. Then we put these estimates together to characterize certain pinched complete ancient solutions and shrinkers in higher codimension. We include some discussion of future research directions in this area of mean curvature flow.